From a conceptual and algorithmic standpoint, I understand how K-means works. However, from a mathematical standpoint, I don't understand why minimizing the WCSS will necessarily maximize the distance between clusters. In other words, can somebody show how this function is equal to maximizing the distance between clusters? It would be helpful to see a derivation that shows all of the steps or please point me to the appropriate reference(s).

Update I found this Witten and Tibshirani reference but it isn't obvious how to get from equation 7 to equation 8.

From a conceptual and algorithmic standpoint, I understand how K-means works. However, from a mathematical standpoint, I don't understand why minimizing the WCSS (within-cluster sums of squares) will necessarily maximize the distance between clusters. In other words, can somebody show how this function is equal to maximizing the distance between clusters? It would be helpful to see a derivation that shows all of the steps or please point me to the appropriate reference(s).

Update I found this Witten and Tibshirani reference but it isn't obvious how to get from equation 7 to equation 8.